Theorem fin23lem30 10029

Description: Lemma for fin23 10076. The residual is disjoint from the common set. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Hypotheses

Ref	Expression
fin23lem.a	⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
fin23lem17.f	⊢ 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔 ↑m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎‘𝑥) → ∩ ran 𝑎 ∈ ran 𝑎)}
fin23lem.b	⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
fin23lem.c	⊢ 𝑄 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ 𝑃 (𝑥 ∩ 𝑃) ≈ 𝑤))
fin23lem.d	⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e	⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))

Assertion

Ref	Expression
fin23lem30	⊢ (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)

Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎

Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem30

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fin23lem.e	. 2 ⊢ 𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
2		eqif 4497	. . 3 ⊢ (𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))))
3	2	biimpi 215	. 2 ⊢ (𝑍 = if(𝑃 ∈ Fin, (𝑡 ∘ 𝑅), ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) → ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))))
4		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun 𝑡)
5		fin23lem.d	. . . . . . . . . . . 12 ⊢ 𝑅 = (𝑤 ∈ ω ↦ (℩𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
6	5	funmpt2 6457	. . . . . . . . . . 11 ⊢ Fun 𝑅
7		funco 6458	. . . . . . . . . . 11 ⊢ ((Fun 𝑡 ∧ Fun 𝑅) → Fun (𝑡 ∘ 𝑅))
8	4, 6, 7	sylancl 585	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → Fun (𝑡 ∘ 𝑅))
9		elunirn 7106	. . . . . . . . . 10 ⊢ (Fun (𝑡 ∘ 𝑅) → (𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ↔ ∃𝑏 ∈ dom (𝑡 ∘ 𝑅)𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏)))
10	8, 9	syl 17	. . . . . . . . 9 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ↔ ∃𝑏 ∈ dom (𝑡 ∘ 𝑅)𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏)))
11		dmcoss 5869	. . . . . . . . . . . 12 ⊢ dom (𝑡 ∘ 𝑅) ⊆ dom 𝑅
12	11	sseli 3913	. . . . . . . . . . 11 ⊢ (𝑏 ∈ dom (𝑡 ∘ 𝑅) → 𝑏 ∈ dom 𝑅)
13		fvco 6848	. . . . . . . . . . . . . . . 16 ⊢ ((Fun 𝑅 ∧ 𝑏 ∈ dom 𝑅) → ((𝑡 ∘ 𝑅)‘𝑏) = (𝑡‘(𝑅‘𝑏)))
14	6, 13	mpan 686	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ dom 𝑅 → ((𝑡 ∘ 𝑅)‘𝑏) = (𝑡‘(𝑅‘𝑏)))
15	14	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡 ∘ 𝑅)‘𝑏) = (𝑡‘(𝑅‘𝑏)))
16	15	eleq2d 2824	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) ↔ 𝑎 ∈ (𝑡‘(𝑅‘𝑏))))
17		incom 4131	. . . . . . . . . . . . . . . 16 ⊢ ((𝑡‘(𝑅‘𝑏)) ∩ ∩ ran 𝑈) = (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏)))
18		difss 4062	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ω ∖ 𝑃) ⊆ ω
19		ominf 8964	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ¬ ω ∈ Fin
20		fin23lem.b	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑃 = {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)}
21	20	ssrab3 4011	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑃 ⊆ ω
22		undif 4412	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
23	21, 22	mpbi 229	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑃 ∪ (ω ∖ 𝑃)) = ω
24		unfi 8917	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
25	23, 24	eqeltrrid 2844	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
26	25	ex 412	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑃 ∈ Fin → ((ω ∖ 𝑃) ∈ Fin → ω ∈ Fin))
27	19, 26	mtoi 198	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑃 ∈ Fin → ¬ (ω ∖ 𝑃) ∈ Fin)
28	27	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (ω ∖ 𝑃) ∈ Fin)
29	5	fin23lem22 10014	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
30	18, 28, 29	sylancr 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
31		f1of 6700	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω⟶(ω ∖ 𝑃))
32	30, 31	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑅:ω⟶(ω ∖ 𝑃))
33		simpr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ dom 𝑅)
34	32	fdmd 6595	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → dom 𝑅 = ω)
35	33, 34	eleqtrd 2841	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → 𝑏 ∈ ω)
36	32, 35	ffvelrnd 6944	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅‘𝑏) ∈ (ω ∖ 𝑃))
37	36	eldifbd 3896	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅‘𝑏) ∈ 𝑃)
38	20	eleq2i 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅‘𝑏) ∈ 𝑃 ↔ (𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)})
39	37, 38	sylnib 327	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ (𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)})
40	36	eldifad 3895	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑅‘𝑏) ∈ ω)
41		fveq2 6756	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = (𝑅‘𝑏) → (𝑡‘𝑣) = (𝑡‘(𝑅‘𝑏)))
42	41	sseq2d 3949	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = (𝑅‘𝑏) → (∩ ran 𝑈 ⊆ (𝑡‘𝑣) ↔ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏))))
43	42	elrab3 3618	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑅‘𝑏) ∈ ω → ((𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} ↔ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏))))
44	40, 43	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑅‘𝑏) ∈ {𝑣 ∈ ω ∣ ∩ ran 𝑈 ⊆ (𝑡‘𝑣)} ↔ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏))))
45	39, 44	mtbid 323	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ¬ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)))
46		fin23lem.a	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡‘𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡‘𝑖) ∩ 𝑢))), ∪ ran 𝑡)
47	46	fin23lem20 10024	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅‘𝑏) ∈ ω → (∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) ∨ (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅))
48	40, 47	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) ∨ (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅))
49		orel1 885	. . . . . . . . . . . . . . . . 17 ⊢ (¬ ∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) → ((∩ ran 𝑈 ⊆ (𝑡‘(𝑅‘𝑏)) ∨ (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅) → (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅))
50	45, 48, 49	sylc 65	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (∩ ran 𝑈 ∩ (𝑡‘(𝑅‘𝑏))) = ∅)
51	17, 50	eqtrid 2790	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ((𝑡‘(𝑅‘𝑏)) ∩ ∩ ran 𝑈) = ∅)
52		disj 4378	. . . . . . . . . . . . . . 15 ⊢ (((𝑡‘(𝑅‘𝑏)) ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ (𝑡‘(𝑅‘𝑏)) ¬ 𝑎 ∈ ∩ ran 𝑈)
53	51, 52	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → ∀𝑎 ∈ (𝑡‘(𝑅‘𝑏)) ¬ 𝑎 ∈ ∩ ran 𝑈)
54		rsp 3129	. . . . . . . . . . . . . 14 ⊢ (∀𝑎 ∈ (𝑡‘(𝑅‘𝑏)) ¬ 𝑎 ∈ ∩ ran 𝑈 → (𝑎 ∈ (𝑡‘(𝑅‘𝑏)) → ¬ 𝑎 ∈ ∩ ran 𝑈))
55	53, 54	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ (𝑡‘(𝑅‘𝑏)) → ¬ 𝑎 ∈ ∩ ran 𝑈))
56	16, 55	sylbid 239	. . . . . . . . . . . 12 ⊢ (((𝑃 ∈ Fin ∧ Fun 𝑡) ∧ 𝑏 ∈ dom 𝑅) → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈))
57	56	ex 412	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom 𝑅 → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈)))
58	12, 57	syl5 34	. . . . . . . . . 10 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑏 ∈ dom (𝑡 ∘ 𝑅) → (𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈)))
59	58	rexlimdv 3211	. . . . . . . . 9 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∃𝑏 ∈ dom (𝑡 ∘ 𝑅)𝑎 ∈ ((𝑡 ∘ 𝑅)‘𝑏) → ¬ 𝑎 ∈ ∩ ran 𝑈))
60	10, 59	sylbid 239	. . . . . . . 8 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) → ¬ 𝑎 ∈ ∩ ran 𝑈))
61	60	ralrimiv 3106	. . . . . . 7 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → ∀𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ¬ 𝑎 ∈ ∩ ran 𝑈)
62		disj 4378	. . . . . . 7 ⊢ ((∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ ∪ ran (𝑡 ∘ 𝑅) ¬ 𝑎 ∈ ∩ ran 𝑈)
63	61, 62	sylibr 233	. . . . . 6 ⊢ ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈) = ∅)
64		rneq 5834	. . . . . . . . 9 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ran 𝑍 = ran (𝑡 ∘ 𝑅))
65	64	unieqd 4850	. . . . . . . 8 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ∪ ran 𝑍 = ∪ ran (𝑡 ∘ 𝑅))
66	65	ineq1d 4142	. . . . . . 7 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = (∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈))
67	66	eqeq1d 2740	. . . . . 6 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ((∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅ ↔ (∪ ran (𝑡 ∘ 𝑅) ∩ ∩ ran 𝑈) = ∅))
68	63, 67	syl5ibr 245	. . . . 5 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → ((𝑃 ∈ Fin ∧ Fun 𝑡) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
69	68	expd 415	. . . 4 ⊢ (𝑍 = (𝑡 ∘ 𝑅) → (𝑃 ∈ Fin → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)))
70	69	impcom 407	. . 3 ⊢ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
71		rneq 5834	. . . . . . . 8 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ran 𝑍 = ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
72	71	unieqd 4850	. . . . . . 7 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → ∪ ran 𝑍 = ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))
73	72	ineq1d 4142	. . . . . 6 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = (∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ∩ ∩ ran 𝑈))
74		rncoss 5870	. . . . . . . 8 ⊢ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
75	74	unissi 4845	. . . . . . 7 ⊢ ∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
76		disj 4378	. . . . . . . 8 ⊢ ((∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∩ ∩ ran 𝑈) = ∅ ↔ ∀𝑎 ∈ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ¬ 𝑎 ∈ ∩ ran 𝑈)
77		eluniab 4851	. . . . . . . . . 10 ⊢ (𝑎 ∈ ∪ {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} ↔ ∃𝑏(𝑎 ∈ 𝑏 ∧ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)))
78		eleq2 2827	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ∈ 𝑏 ↔ 𝑎 ∈ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)))
79		eldifn 4058	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → ¬ 𝑎 ∈ ∩ ran 𝑈)
80	78, 79	syl6bi 252	. . . . . . . . . . . . 13 ⊢ (𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ∈ 𝑏 → ¬ 𝑎 ∈ ∩ ran 𝑈))
81	80	rexlimivw 3210	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈) → (𝑎 ∈ 𝑏 → ¬ 𝑎 ∈ ∩ ran 𝑈))
82	81	impcom 407	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝑏 ∧ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → ¬ 𝑎 ∈ ∩ ran 𝑈)
83	82	exlimiv 1934	. . . . . . . . . 10 ⊢ (∃𝑏(𝑎 ∈ 𝑏 ∧ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → ¬ 𝑎 ∈ ∩ ran 𝑈)
84	77, 83	sylbi 216	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)} → ¬ 𝑎 ∈ ∩ ran 𝑈)
85		eqid 2738	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈))
86	85	rnmpt 5853	. . . . . . . . . 10 ⊢ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)}
87	86	unieqi 4849	. . . . . . . . 9 ⊢ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) = ∪ {𝑏 ∣ ∃𝑧 ∈ 𝑃 𝑏 = ((𝑡‘𝑧) ∖ ∩ ran 𝑈)}
88	84, 87	eleq2s 2857	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) → ¬ 𝑎 ∈ ∩ ran 𝑈)
89	76, 88	mprgbir 3078	. . . . . . 7 ⊢ (∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∩ ∩ ran 𝑈) = ∅
90		ssdisj 4390	. . . . . . 7 ⊢ ((∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ⊆ ∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∧ (∪ ran (𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∩ ∩ ran 𝑈) = ∅) → (∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ∩ ∩ ran 𝑈) = ∅)
91	75, 89, 90	mp2an 688	. . . . . 6 ⊢ (∪ ran ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) ∩ ∩ ran 𝑈) = ∅
92	73, 91	eqtrdi 2795	. . . . 5 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)
93	92	a1d 25	. . . 4 ⊢ (𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
94	93	adantl 481	. . 3 ⊢ ((¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄)) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
95	70, 94	jaoi 853	. 2 ⊢ (((𝑃 ∈ Fin ∧ 𝑍 = (𝑡 ∘ 𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧 ∈ 𝑃 ↦ ((𝑡‘𝑧) ∖ ∩ ran 𝑈)) ∘ 𝑄))) → (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅))
96	1, 3, 95	mp2b 10	1 ⊢ (Fun 𝑡 → (∪ ran 𝑍 ∩ ∩ ran 𝑈) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ifcif 4456  𝒫 cpw 4530  ∪ cuni 4836  ∩ cint 4876   class class class wbr 5070   ↦ cmpt 5153  dom cdm 5580  ran crn 5581   ∘ ccom 5584  suc csuc 6253  Fun wfun 6412  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  ℩crio 7211  (class class class)co 7255   ∈ cmpo 7257  ωcom 7687  seq_ωcseqom 8248   ↑_m cmap 8573   ≈ cen 8688  Fincfn 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-seqom 8249  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628

This theorem is referenced by:  fin23lem31  10030